HINT: <no title>
[−0 points ⇒ 4 / 4 points left]
The coordinates given look a bit complex, but focus
on what this question is asking about. Start by writing down the formula
for what you need to calculate, and substitute the coordinates into it.
STEP: Substitute the values into the distance formula
[−1 point ⇒ 3 / 4 points left]
This is a distance formula question: we need to calculate the distance between the points (3;−3) and (53‾√;53‾√). On a graph, the points and the distance that we want look like this:
To begin, we need to substitute the coordinates into
the formula. It can be helpful to list the coordinates from the points
according to the values we need in the distance formula.
x1x2=3=53‾√;y1;y2=−3=53‾√
Now substitute these values into the formula. Use
brackets carefully here to keep everything organised. (There will be
lots of brackets, so we will change the brackets in the formula into
square brackets to distinguish them from the substituted values.)
d=[x2−x1]2+[y2−y1]2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=[(53‾√)−(3)]2+[(53‾√)−(−3)]2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√
STEP: Simplify the square brackets and square them
[−2 points ⇒ 1 / 4 points left]
Now we follow BODMAS, which means we need to
simplify inside the square brackets as much as possible. However, do not
change the surds into decimals: often expressions like this simplify
after a few steps. Let's see what happens when we expand the square
brackets.
Notice that there is a double negative in the second square bracket which changes to addition.
d=[53‾√−3]2+[53‾√+3]2‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=[53‾√−3][53‾√−3]+[53‾√+3][53‾√+3]‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=(75−303‾√+9)+(75+303‾√+9)‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√=75+9+75+9‾‾‾‾‾‾‾‾‾‾‾‾‾‾‾√
That simplified a lot! The key part is the cancellation of the surd terms, −303‾√ and 303‾√.
From this point on there are only integer values to work with inside the root.
STEP: Complete the calculation
[−1 point ⇒ 0 / 4 points left]
Now we can finish the calculation. Remember, we
cannot evaluate the square root until all of the calculations inside of
it are done, because a root acts like brackets on the expression under
it.
d=84+84‾‾‾‾‾‾‾√=168‾‾‾‾√
In this case, the expression cannot be simplified
any more: it is a surd. The question tells us the answer must exact, so
we have the answer. (This surd can be simplied to 242‾‾‾√, which is also accepted.)
The distance between the points is: 168‾‾‾‾√.
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